Modeling with Equations A population that experiences exponential growth increases according to the model n(t)=n0ert where: n(t) = population at time t n0 = initial size of the population r = relative rate of growth t = time

Ex 1: The initial bacterium count in a culture is 500 Ex 1: The initial bacterium count in a culture is 500. A biologist later makes a sample count of bacteria in the culture and finds that the relative rate of growth is 40% per hour. a. Find a function that models the number of bacteria after t hours. b. What is the estimated count after 10 hours?

Ex 2: In 2000 the population of the world was 6. 1 Ex 2: In 2000 the population of the world was 6.1 billion and the relative rate of growth was 1.4% per year. It is claimed that a rate of 1.0% per year would make a significant difference in the total population in just a few decades. Test this claim by estimating the population of the world in the year 2050 using a relative rate of growth of (a) 1.4% per year and (b) 1.0% per year.

Ex 3: A certain breed of rabbit was introduced onto a Ex 3: A certain breed of rabbit was introduced onto a small island about 8 years ago. The current rabbit population on the island is estimated to be 4100, with a relative growth rate of 55% per year. a. What was the initial size of the rabbit population? b. Estimate the population 12 years from now.

Ex 4: The population of the world in 2000 was 6. 1 Ex 4: The population of the world in 2000 was 6.1 billion, and the estimated relative growth rate was 1.4% per year. If the population continues to grow at this rate, when will it reach 122 billion?

Ex 5: A culture starts with 10,000 bacteria, and the Ex 5: A culture starts with 10,000 bacteria, and the number doubles every 40 minutes. a. Find a function that models the number of bacteria at time t. b. Find the number of bacteria after 1 hour. c. After how many minutes will there be 50,000 bacteria?

Radioactive Decay Model If m0 is the initial mass of a radioactive substance with half-life h, then the mass remaining at time t is modeled by the function m(t)=m0e-rt where:

Ex 6: Polonium-210 has a half-life of 140 days Ex 6: Polonium-210 has a half-life of 140 days. Suppose a sample of this substance has a mass of 300 mg. a. Find a function that models the amount of the sample remaining at time t. b. Find the mass remaining after one year. c. How long will it take for the sample to decay to a mass of 200 mg?

Newton’s Law of Cooling If D0 is the initial temperature difference between an object and its surroundings, and if its surroundings have temperature TS, then the temperature of the object at time t is modeled by the function T(t)=TS + D0e-kt where k is a positive constant that depends on the type of object.

Ex 7: A cup of coffee has a temperature of 200°F and Ex 7: A cup of coffee has a temperature of 200°F and is placed in a room that has a temperature of 70°F. After 10 min the temperature of the coffee is 150°F. a. Find a function that models the temperature of the coffee at time t. b. Find the temperature of the coffee after 15 min. c. When will the coffee have cooled to 100°F?

Logarithmic Scales Ex 8: pH = -log[H+] a. The hydrogen ion concentration of a sample of human blood was measured to be [H+] = 3.16 x 10-8 M. Find the pH and classify the blood as acidic or basic. b. The most acidic rainfall ever measured occurred in Scotland in 1974; its pH was 2.4. find the hydrogen ion concentration.

Ex 9: The 1906 earthquake in San Francisco had an estimated magnitude of 8.3 on the Richter scale. In the same year a powerful earthquake occurred on the Columbia-Ecuador border and was four times as intense. What was the magnitude of the Columbia-Ecuador earthquake on the Richter scale?

Ex 10: The 1989 Loma Prieta earthquake that shook San Francisco had a magnitude of 7.1 on the Richter scale. How many times more intense was the 1906 earthquake that had a magnitude of 8.3 on the Richter scale?

Ex 11: Find the decibel intensity level of a jet engine during takeoff if the intensity was measured at 100 W/m2 (I0 = 10-12).

Assignment (#32) S 5.5: pg 438 – 439 #2,8,14,15,23,27,36